Theorem sseqtrrd 3223

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2202	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3222	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  sseqtrrid  3235  fnfvima  5800  tfrlemiubacc  6397  tfr1onlemubacc  6413  tfrcllemubacc  6426  rdgivallem  6448  nnnninf  7201  nninfwlpoimlemg  7250  dfphi2  12413  ctinf  12672  imasaddfnlemg  13016  imasaddvallemg  13017  subsubm  13185  subsubg  13403  subsubrng  13846  subsubrg  13877  lidlss  14108  toponss  14346  ssntr  14442  iscnp3  14523  cnprcl2k  14526  tgcn  14528  tgcnp  14529  ssidcn  14530  cncnp  14550  txcnp  14591  imasnopn  14619  hmeontr  14633  blssec  14758  blssopn  14805  xmettx  14830  metcnp  14832  plyaddlem1  15067  plymullem1  15068  plycoeid3  15077  nnsf  15736  nninfsellemsuc  15743